Design, analysis, and feedback control of a nonlinear micro-piezoelectric–electrostatic energy harvester

Abstract

A nonlinear micro-piezoelectric–electrostatic energy harvester is designed and studied using mathematical and computational methods. The system consists of a cantilever beam substrate, a bimorph piezoelectric transducer, a pair of tuning parallel-plate capacitors, and a tip–mass. The governing nonlinear mathematical model of the electro-mechanical system including nonlinear material and quadratic air-damping is derived for the series connection of the piezoelectric layers. The static and modal frequency curves are computed to optimize the operating point, and a parametric study is performed using numerical methods. A bias DC voltage is used to adapt the system to resonate with respect to the frequency of external vibration. Furthermore, to improve the bandwidth and performance of the harvester (and achieve a high level of harvested power without sacrificing the bandwidth), a nonlinear feedback loop is integrated into the design.

Appendix: Parameter expressions

In the following equations, prime $(\,)'$ represents the spatial derivative,

$$\begin{aligned} k_l&= E_s I_s \int _0^{L_s} \psi ''(r){}^2 \, \hbox {d}r \end{aligned}$$

(26)

$$\begin{aligned} k_n&= E_s I_s \int _0^{L_s} \psi '(r){}^2 \psi ''(r){}^2 \, \hbox {d}r \end{aligned}$$

(27)

$$\begin{aligned} m_l&= m_p \int _0^{L_p} \psi (r)^2 \, dr + m_s \int _0^{L_s} \psi (r)^2 \hbox {d}r \nonumber \\&\quad + m_t \psi \left( L_s\right) ^2 + J_t \psi '\left( L_s\right) ^2 \end{aligned}$$

(28)

$$\begin{aligned} m_n&= m_t \left( \int _0^{L_s} \psi '(r)^2 \, \hbox {d}r \right) ^2 + J_t \psi '(L_s)^4 \nonumber \\&\quad + m_p \int _0^{L_p} \left( \int _0^r \psi '(r)^2 \, dr\right) ^2 \hbox {d}r \nonumber \\&\quad + m_s \int _0^{L_s} \left( \int _0^r \psi '(r)^2 \hbox {d}r\right) ^2 \hbox {d}r \end{aligned}$$

(29)

$$\begin{aligned} f_l&= m_p \int _0^{L_p} \psi (r) z(r) \, dr + m_s \int _0^{L_s} \psi (r) z(r) \hbox {d}r \nonumber \\&\quad + m_t \psi \left( L_s\right) z\left( L_s\right) \end{aligned}$$

(30)

$$\begin{aligned} f_n&= m_t z(L_s)^2 + m_p \int _0^{L_p} z(r)^2 \hbox {d}r \nonumber \\&\quad + m_s \int _0^{L_s} z(r)^2 \hbox {d}r \end{aligned}$$

(31)

$$\begin{aligned} h^q_l&= \frac{1}{12} b c_p^{11} h_p \left( 4 h_p ^2+6 h_p h_s +3 h_s ^2\right) \int _0^{ L_p } \psi ''(r)^2 \, \hbox {d}r \end{aligned}$$

(32)

$$\begin{aligned} h^\lambda _l&= -\frac{1}{2} b e_{31} ( h_p + h_s ) \int _0^{ L_p } \psi ''(r) \hbox {d}r \end{aligned}$$

(33)

$$\begin{aligned} h^\lambda _n&= -\frac{1}{4} b e_{31} ( h_p + h_s ) \int _0^{ L_p } \psi '(r)^2 \psi ''(r) \hbox {d}r \nonumber \\&\quad -\frac{1}{24} b e_{3111} \left( 2 h_p ^3+4 h_p ^2 h_s +3 h_p h_s ^2+ h_s ^3\right) \nonumber \\&\quad \times \int _0^{ L_p } \psi ''(r)^3 \hbox {d}r \end{aligned}$$

(34)

$$\begin{aligned} h^q_n&= \frac{1}{12} b c_p^{11} h_p \left( 4 h_p^2 + 6 h_p h_s +3 h_s^2 \right) \int _0^{ L_p } \psi '(r)^2 \psi ''(r)^2 \hbox {d}r \nonumber \\&\quad +\frac{1}{160} b c_p^{1111} h_p \nonumber \\&\quad \times \left( 16 h_p^4 + 40 h_p^3 h_s + 40 h_p^2 h_s^2 + 20 h_p h_s ^3 + 5 h_s ^4 \right) \nonumber \\&\quad \times \int _0^{ L_p } \psi ''(r)^4 \hbox {d}r \end{aligned}$$

(35)

$$\begin{aligned} C_p&= -\frac{b L_p \epsilon _{33}}{4 h_p } \end{aligned}$$

(36)

$$\begin{aligned} \gamma _l&= \frac{\epsilon _0 \epsilon _r}{g^3} \left( 2 b_s + 0.33125 \left( b_s^{0.25} + h_s^{0.25} \right) g^{0.75} \right) \nonumber \\&\quad \times \int _{0}^{L_m} (\psi (L_s) + r \psi '(L_s)) ^2 \, \hbox {d}r \end{aligned}$$

(37)

$$\begin{aligned} \gamma _n&= \frac{\epsilon _0 \epsilon _r}{g^5} \left( 2 b_s + 0.201856 \left( b_s^{0.25} + h_s^{0.25} \right) g^{0.75} \right) \nonumber \\&\quad \times \int _{0}^{L_m} (\psi (L_s) + r \psi '(L_s)) ^4 \, \hbox {d}r \end{aligned}$$

(38)